Peterson got his wish, and this time he couldn't deny Jones got the best of him.
Jones had 10 catches for a career-high 189 yards and a touchdown for the Falcons (5-7), the NFC South leaders.
The Cardinals (9-3) have lost two straight, cutting their lead to one game over Seattle in the NFC West.
"It's going to be a long plane ride home," Peterson said.
Peterson played for LSU, Jones played for Alabama, and the rivalry that started in college has continued in the NFL.
"He won a couple battles, and I won a couple battles, but for the most part I believe I won the majority of those battles," Peterson had said on Wednesday.
That boast caught Jones' attention.
"Supposedly, Pat said he won most of the battles between me and him," Jones said, adding he made sure he had "proper rest" to ensure "everything was down" in his timing with quarterback Matt Ryan.
"I am pretty sure it was a statement game for Julio," Atlanta's Devin Hester said. "He told me that every night, and every day he thought about it, so I do not know if it stirred him up, but he showed it on the field."
Cardinals coach Bruce Arians clearly wasn't happy Peterson couldn't back up his boast.
"You better watch out when you call people out," Arians said after the game. "You better be ready to play. If you want to call him out, call him out, but don't give up ... a career day."
Peterson said Jones "won the matchup today" and added, "I mean, Cy Young winners give up home runs."
The Falcons took a 17-0 lead in the first quarter.
Steven Jackson's 55-yard run — his longest run since 2009 — set up Ryan's 1-yard touchdown pass to Levine Toilolo. Ryan threw a 32-yard scoring pass to Jones later in the quarter.
Ryan was 30 of 41 for 361 yards and a pair of TDs.
Arizona's only touchdown until late in the game came on an 88-yard interception return by safety Rashad Johnson in the second quarter.
The Cardinals went more than 11 quarters without an offensive touchdown until Drew Stanton's 3-yard scoring pass to running back Stepfan Taylor with just over a minute remaining.
Here are some things to know about the Falcons' first win over a team with a winning record:
CALL IT BACK: Hester had an apparent 70-yard punt return for a touchdown called back when he was flagged for an offensive facemask penalty as he threw a stiff-arm on Cardinals punter Drew Butler. The replays showed Butler grabbed Hester's facemask.
Falcons special teams coach Keith Armstrong received an unsportsmanlike conduct penalty for protesting the call.
CARDINALS HURTING: The Cardinals were without receiver Larry Fitzgerald, who missed his second straight game with a strained left knee. Running back Andre Ellington didn't return after leaving with a hip injury in the first half.
JACKSON HITS 100: Jackson had 18 carries for 101 yards, his first 100-yard game since Nov. 25, 2012, for the Rams at Arizona. "Just another day at the office," he said.
CAREER DAY FOR BRYANT: Matt Bryant kicked a career-best five field goals without a miss for Atlanta. He was good from 20, 23, 21, 45 and 28 yards.
"In a perfect world, you hope you're kicking 17 extra points, but if they count on you to get a field goal, I just go out there and do it," he said.
NO OFFENSE: Stanton completed 24 of 39 passes for 294 yards with one touchdown and two interceptions. The Cardinals were successful on only 1 of 7 third-down plays.
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